TECHNICAL

42

Steel Construction Vol. 32 No. 1 February 2008

Last year the SAISC took its “Design of light industrial buildings” course to a

number of cities and towns throughout South Africa. One of the things that

became apparent to me while lecturing on the course was that a number of

design engineers do not fully appreciate or understand the requirements of

clause 13.3.2 of the current South African hot-rolled steel design code.

Consequently, they are in the dark with regards to the design of asymmetric and

singly symmetric sections. This article, dealing with the basics of concentric

elastic buckling of columns, is the first in a series to address the design of these

types of sections.

My guess is that most structural engineers (myself included) grew up on a diet of

elastic flexural buckling. Euler’s celebrated formula was drummed into our heads

from early on in our academic careers. At an undergraduate level, torsional and

flexural-torsional buckling were topics that were possibly not covered in much

depth or were topics that were way too strenuous to contemplate with heavy eye-

lids. When the first limit states design code for structural steelwork

(SABS0162-1:1993) was introduced in South Africa, it lacked guidance on how to

design singly symmetric or asymmetric compression members. To address this state

of affairs, the SAISC published a technical note in Steel Construction

(November/December 1993) which contained a proposed methodology that has

now essentially been incorporated into the current design code as clause 13.2.2.

Those engineers who are familiar with the South African limit states design code

for cold-formed steelwork would have recognised the singly symmetric elastic

buckling equations since they are similar to those given in SANS10162-2:1993.

Other than nomenclature, the main difference is that the cold-formed code implic-

itly considers the section to be symmetrical about the x axis (as opposed to the

y-axis in 10162-1). The cubic equation for elastic buckling of asymmetric sections

was, however, unfamiliar territory for many engineers.

Understanding the nature of elastic buckling of columns, be it flexural, torsional

or flexural-torsional, can be enhanced by unpacking the equation given in

clause 13.3.2 c) viz:

The derivation of this equation can be found in many classic mechanics text-

book. So let’s not get into that level of detail here except to point out that it is

generally expressed in terms of elastic buckling loads and not stresses.

The solution to this equation gives the elastic buckling stress, f

e

, of a column.

The terms f

ex

, f

ey

and f

ez

refer to the elastic buckling stress of the column about

it’s strong axis, about it’s weak axis and the elastic torsional buckling stress

respectively. The polar moment of inertia of the centroid about the shear centre

is denoted by r

o

. The terms x

o

and y

o

are defined as the principal coordinates of

the shear centre with respect to the centroid of the cross-section. To clarify

what this means, consider diagram 1 (shown left) of a general open cross-

section. The axes labelled X and Y are the principal centroidal axes of the

section, point C identifies the location of the centroid and point O the shear

centre. x

o

and y

o

are the dimensions as shown.

It is a well know fact that if the cross-section of a column has two axes of

symmetry, then the shear centre coincides with the centroid of the cross-

Equation 1.

Diagram 1.

ASYMMETRIC

AND SINGLY

SYMMETRIC

SECTIONS IN

COMPRESSION

By David Blitenthall, development

engineer, SAISC

My guess is that most structural

engineers (myself included) grew up

on a diet of elastic flexural buckling.

Euler’s celebrated formula was

drummed into our heads from early

on in our academic careers. At an

undergraduate level, torsional and

flexural-torsional buckling were

topics that were possibly not covered

in much depth or were topics that

were way too strenuous to

contemplate with heavy eye-lids.

TECHNICAL

Steel Construction Vol. 32 No. 1 February 2008

43

Equation 3.

Equation 2.

section. In this case x

o

= y

o

= 0 mm. Substituting these values into equation 1,

gives the following:

The three solutions to this equation are obviously f

e

= f

ex

or f

ey

or f

ez

. The

column will therefore buckle at the lowest of these stresses and in a correspon-

ding mode viz. buckling either occurs by flexure about the strong axis, flexure

about the weak axis or by torsion. It is also important to note that these three

modes are independent in this case.

For any section having only one axis of symmetry, the shear centre will be on

that axis but generally not at the centroid. In the case where the y axis is the

axis of symmetry (a T-section for example), x

o

= 0 mm. Substituting this into

equation (1), give the following:

Substituting Ω = 1 – (y

o

/ r

o

)

2

and solving the term in square parenthesis using

the quadratic formula, the solutions to this equation are f

e

= f

ex

and

.

Such a column will therefore buckle at the lowest of the three stresses,

either in a flexural mode about the x-axis at f

ex

or in a torsional-flexural

mode at the lowest root given by the quadratic equation – this root being

lower than either f

ey

or f

ez

. In this case, flexural buckling about the x-x axis

is independent while flexural buckling about the y-y axis and torsional buck-

ling are coupled. In the case where the x axis is the axis of symmetry (a

channel section for example), y

o

= 0mm and the solution to equation (1) can

be found by interchanging the y and x terms such that f

e

= f

ey

and

.

For an asymmetric section, x

o

Ω0 and y

o

Ω0 and the solution of f

e

requires

finding the lowest root of equation (1). The buckling modes in this instance

are interdependent and the lowest buckling mode is always less than either

f

ex

, f

ey

or f

ez

.

The next article on this subject will take a look at how this basic elastic buckling

theory is applied in SANS10162-1.

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